What number completes the sequence?

[Zetep]

Shown below is a 4 by 4 matrix:

999 998 997 996
729 648 567 486
126 192 210 192
12 18 ? 18

So far, I've not come up with the 'missing' number, but in the process of attempting to develop some sort of 'magic' solution, I've found some interesting relationships with the other numbers.

For example, the average of a(1,2) and a(1,4), 998 and 996, is the a(1,3) number 997. Then, the average of a(2,2) and a(2,4), 648 and 486, is the C(2,3) number 567.

There seems to be other relationships, i.e., a(3,2) plus a(4,2) equals a(3,3). So does
a(3,4) plus a(4,4) equal a(3,3). Plus other similar associations.

Evidently, my once keen mind is not so keen these days with being able to solve this 'mysterious' matrix. Any counsel, advice and/or assistance would be seriously appreciated.

Zetep

 

[james1234567890]

The missing number is 0. Each number in the matrix is the product of the digits of the number in the previous row but the same column of the matrix. So the missing number (4,3) is product of the digits of (3,3) or 210 = 2*1*0 = 0.

 

[Zetep]

Found The Missing Number

The missing number is 0. Each number in the matrix is the product of the digits of the number in the previous row but the same column of the matrix. So the missing number (4,3) is product of the digits of (3,3) or 210 = 2*1*0 = 0.

Thanks for your reply ... very much appreciated. Obviously, the solution is straightforward when the method you described is taken. I'm curious to know how it was that you chose to apply your approach. Were there 'clues' given the matrix numbers that appeared visible to your keen eye?

Zetep

 

[james1234567890]

Thanks a lot for your appreciation. This idea struck me when I looked at the first two numbers in the first column i.e. 999 and 729. I remembered that 729 is 9 cubed i.e. 9 * 9 * 9. To my surprise, the number below also follwed the same convention i.e. 126 = 7 * 2 * 9. Thereafter I tried to apply this rule to all numbers in the martrix and it turned out to be correct.